Del Pezzo elliptic varieties of degree d <=4

Abstract

Let Y be a smooth del Pezzo variety of dimension n ≥ 3, i.e. a smooth complex projective variety endowed with an ample divisor H such that −KY = (n − 1)H. Let d be the degree Hn of Y and assume that d ≤ 4. Consider a linear subsystem of |H| whose base locus is zero-dimensional of length d. The subsystem defines a rational map onto Pn−1 and, under some mild extra hypothesis, the general pseudofibers are elliptic curves. We study the elliptic fibration X → Pn−1 obtained by resolving the indeterminacy and call the variety X a del Pezzo elliptic variety. Extending the results of [7] we mainly prove that the Mordell-Weil group of the fibration is finite if and only if the Cox ring of X is finitely generated. Mathematics Subject Classification (2010): 14C20 (primary); 14Q15, 14E05, 14N25 (secondary).